What is the average of all positive integers that have four digits when written in base 3, but two digits when written in base 6? Write your answer in base 10.
Smallest two digit base 6 number 106 = 610
Largest two digit base 6 5 5 = 3510
Largest base 3 number w 4 digits 22223 = 80
smallest 1000 = 27
so numbers from 27 to 35 are in the set of parameters asked in the Q
(27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 ) / 9 = .............
+285 base 3 numbers with 4 digits - 1000-2222 which is 27-80
base 6 numbers with 2 digits- 10-55 which is 6-35
the range is 27-35
the average is just the middle number for odd number consecutive positives
middle is 31
So the average of the positive integers, when written in base 3 that contains 4 digits but in base 6 contain 2 digits, is 31.
